A Superposition Solution of Heat Conduction in a Cavitied Region Subjected to a Convective Boundary Condition

Abstract

The problem of heat conduction in an irregular region subjected to a convective boundary condition can not be solved by the classical conformal mapping or the Schwarz iteration method. In this paper, a superposition technique is attempted. The problem solved consists of a two-dimensional Cartesian system with temperature specified on the irregular boundary and a convective condition appearing on the regular surface, other surfaces being adiabatic. In the solution, the original problem with an irregular lower boundary is replaced by two subproblems with regular boundaries. The general solution derived in this paper is specialized to four cavity geometries that include rectangular, elliptical, circular and triangular configurations. The analytical results show that, by using an effective conduction thickness, a better agreement can be reached between the predicted and the true temperatures. This effective thickness can be expressed in terms of the cavity cross-sectional area and the system width and be formulated into a simple formula. The method gives an approximate solution useful to estimate the surface temperature for systems having rectangular, medium-sized cavities.